Jacobi spectral Galerkin method for elliptic Neumann problems

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.     

GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method

A Helmholtz equation in two dimensions discretized by a second order finite difference scheme is considered. Krylov methods such as Bi-CGSTAB and IDR(s) have been chosen as solvers. Since the convergence of the Krylov solvers deteriorates with increasing wave number, a shifted Laplace multigrid preconditioner is used to improve the convergence. The implementation of the preconditioned solver on CPU (Central Processing Unit) is compared to an implementation on GPU (Graphics Processing Units or graphics card) using CUDA (Compute Unified Device Architecture). The results show that preconditioned Bi-CGSTAB on GPU as well as preconditioned IDR(s) on GPU is about 30 times faster than on CPU for the same stopping criterion.     

A fast collocation method for acoustic scattering in shallow oceans

In this paper, we investigate the numerical solution of the integral equation of the second kind reduced by acoustic scattering in shallow oceans with Dirichlet condition. Based on analyzing the singularity of the truncating kernel with a sum of infinite series, using our trigonometric interpolatory wavelets and collocation method, we obtain the numerical solution which possesses a fast convergence rate like o(2^−j). Moreover, the entries of the stiffness matrix can be obtained by FFT, which lead the computational complexity to decrease obviously.     

Application of the discrete Fourier transform to solving the Cauchy integral equation

The uniquely solvable system of the Cauchy integral equation of the first kind and index 1 and an additional integral condition is treated. Such a system arises, for example, when solving the skew derivative problem for the Laplace equation outside an open arc in a plane. This problem models the electric current from a thin electrode in a semiconductor film placed in a magnetic field. A fast and accurate numerical method based on the discrete Fourier transform is proposed. Some computational tests are given. It is shown that the convergence is close to exponential.     

Fourier Moment Method with Regularization for the Cauchy Problem of Helmholtz Equation

In this paper, we consider the reconstruction of the wave field in abounded domain. By choosing a special family of functions, the Cauchy problemcan be transformed into a Fourier moment problem. This problem is ill-posed. Wepropose a regularization method for obtaining an approximate solution to the wavefield on the unspecified boundary. We also give the convergence analysis and errorestimate of the numerical algorithm. Finally, we present some numerical examples toshow the effectiveness of this method.     

Approximation of the Singularities of a Bounded Function by the Partial Sums of Its Differentiated Fourier Series

In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those formulas.     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs

In this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov-Kuznetsov (ZK) equation and the Kadomtsev-Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method.     

求解椭圆方程柯西问题的修正吉洪诺夫正则化方法

研究了一类变系数椭圆方程的柯西问题,这类问题出现在很多实际问题领域.由于问题的不适定性,不可能通过经典的数值方法来求解上述问题,必须引入正则化手段.采用了一种修正吉洪诺夫正则化方法来求解上述问题.在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步显示方法是稳定有效的.     

广义Benjamin-Bona-Mahony方程Fourier谱方法的大时间误差估计

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｅｎｊａｍｉｎ Ｂｏｎａ Ｍａｈｏｎｙｅｑｕａｔｉｏｎｕｔ+ｕｘ+ｕｕｘ -ｕｘｘ-ｕｘｘｔ =0 ( 1 .1 )ｉｎｃｏｒｐｏｒａｔｅｓｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅａｎｄｄｉｓｓｉｐａｔｉｖｅｅｆｆｅｃｔｓ ,ａｎｄｈａｓｂｅｅｎｐｒｏｐｏｓｅｄａｓａｍｏｄｅｌｆｏｒｂｏｔｈｔｈｅｂｏｒｅｐｒｏｐａｇａｔｉｏｎａｎｄｔｈｅｗａｔｅｒｗａｖｅｓ[1,2 ] .Ｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｕｎｉｑｕｅｎｅｓｓｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｉｓｅｑｕａｔｉｏｎｈａｖｅｂｅｅ…     

On a Construction of Fast Direct Solvers

Fast direct solvers for the Poisson equation with homogeneous Dirichlet and Neumann boundary conditions on special triangles and tetrahedra are constructed. The domain given is extended by symmetrization or skew symmetrization onto a rectangle or a rectangular parallelepiped and a fast direct solver is used there. All extendable domains are found. Eigenproblems are also considered.     

Improvement of some bounds on the stability of fast Helmholtz solvers

Three fast Helmholtz solvers based on the Fast Fourier Transform are studied with respect to their stability. It is proved that they are strongly stable, and that the backward relative errors can grow at most like O(log₂ n)₀, where ₀ is the machine roundoff unit.     

Fundamental frequency of a doubly connected membrane: a modified perturbation method

The fundamental frequency of a membrane is the square root ofthe lowest eigenvalue of the negative Laplace operator withDirichlet boundary conditions. A doubly connected membrane withthe inner region of vanishing maximal dimension 2c is consideredin this paper. A modified perturbation method is developed toprovide an asymptotic expansion (c 0) for the fundamental frequencyof the membrane. The first three order terms of the asymptoticexpansion for the fundamental frequency of a doubly connectedmembrane with the circular inner region are derived explicitly.The results are compared with the exact solutions and the approximationsdetermined by other investigators. The error of the perturbationcalculations compared with the exact values is less than 1%as c is less than or equal to 0·25 and is less than 4%as c is less than or equal to 0·35.     

Toward a modified variational iteration method

The variational iteration method (VIM) attracted much attention in the past few years as a promising method for solving nonlinear differential equations. It is shown in this paper that the application of VIM to a special kind of nonlinear differential equations leads to calculation of unneeded terms and more time consumed in repeated calculations for series solutions. A modified VIM is introduced to eliminate the shortcomings; and its effectiveness is illustrated by some examples.     

On a fast and accurate method for computing Fourier transforms

In this paper, a variable order method for the fast and accurate computation of the Fourier transform is presented. The increase in accuracy is achieved by applying corrections to the trapezoidal sum approximations obtained by the FFT method. It is shown that the additional computational work involved is of orderK(2m+2), wherem is a small integer andKn. Analytical expressions for the associated error is also given.     

On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation

In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

On exact solutions to partial differential equations by the modified homotopy perturbation method

Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the effciency of the proposed method.     

A Second-Order Method for the Electromagnetic Scattering from a Large Cavity

In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media.     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.